Linear viscoelasticity relaxation modulus

Stress relaxation for step squeezing of polystyrene at 180°C. (a) Stress versus time for increasing strain steps. Stress increases at short times, 2-10 ms because the plates take a finite time to close. The horizontal stress response signifies transducer overload. The rapid drop for strains e > 1 indicates loss of lubricant, (b) Stress relaxation data plotted as relaxation modulis. Solid line is the linear viscoelastic relaxation modulus calculated from shear dynamic data. Adapted from Soskey and Winter (1985). 

The Maxwell model is also called Maxwell fluid model. Briefly it is a mechanical model for simple linear viscoelastic behavior that consists of a spring of Young s modulus (E) in series with a dashpot of coefficient of viscosity (ji). It is an isostress model (with stress 5), the strain (f) being the sum of the individual strains in the spring and dashpot. This leads to a differential representation of linear viscoelasticity as d /dt = (l/E)d5/dt + (5/Jl)-This model is useful for the representation of stress relaxation and creep with Newtonian flow analysis. 

The linear viscoelastic behavior of liquid and solid materials in general is often defined by the relaxation time spectrum 11(1) [10], which will be abbreviated as spectrum in the following. The transient part of the relaxation modulus as used above is the Laplace transform of the relaxation time spectrum H(l)... 

The time-dependent rheological behavior of liquids and solids in general is described by the classical framework of linear viscoelasticity [10,54], The stress tensor t may be expressed in terms of the relaxation modulus G(t) and the strain history ... 

Here m is the usual small-strain tensile stress-relaxation modulus as described and observed in linear viscoelastic response [i.e., the same E(l) as that discussed up to this point in the chapter). The nonlinearity function describes the shape of the isochronal stress-strain curve. It is a simple function of A, which, however, depends on the type of deformation. Thus for uniaxial extension,... 

The linear viscoelastic properties are often expressed in terms of an auxiliary function, the relaxation time distribution, H(x) H(x)dlnx is the portion of the initial modulus contributed by processes with relaxation times in the range lnt, InT + dlnt ... 

In Eq. (4.13) NT is the total number of internal degrees of freedom per unit volume which relax by simple diffusion (NT — 3vN for dilute solutions), and t, is the relaxation time of the ith normal mode (/ = 1,2,3NT) for small disturbances. Equation (4.13), together with a stipulation that all relaxation times have the same temperature coefficient, provides, in fact, the molecular basis of time-temperature superposition in linear viscoelasticity. It also reduces to the expression for the equilibrium shear modulus in the kinetic theory of rubber elasticity when tj = oo for some of the modes. 

The most obvious problem of non-linearity is the definition of a modulus. For a linear viscoelastic material we need to define not only a real and an imaginary modulus but also a spectrum of relaxation times if we are fully to describe the material - although it is more usual to quote either an isochronous modulus or a modulus at a fixed frequency. We must, for a full description of a non-linear material give the moduli (and relaxation times) as a function of strain as well this will not usually be practicable so we satisfy ourselves by quoting the modulus at a given strain. The question then arises as to whether this... 

Consider two experiments carried out with identical samples of a viscoelastic material. In experiment (a) the sample is subjected to a stress CT for a time t. The resulting strain at f is ei, and the creep compliance measured at that time is D t) = e la. ln experiment (b) a sample is stressed to a level CT2 such that strain i is achieved immediately. The stress is then gradually decreased so that the strain remains at f for time t (i.e., the sample does not creep further). The stress on the material at time t will be a-i, and the corresponding relaxation modulus will be y 2(t) = CT3/C1. In measurements of this type, it can be expected that az> 0 > ct and Y t) (D(r)) , as indicated in Eq. (11-14). G(t) and Y t) are obtained directly only from stress relaxation measurements, while D(t) and J(t) require creep experiments for their direct observation. Tliese various parameters can be related in the linear viscoelastic region described in Section 11.5.2. 

When reptation is used to develop a description of the linear viscoelasticity of polymer melts [5, 6], the same underlying hypothesis ismade, and the same phenomenological parameter Ng appears. Basically, to describe the relaxation after a step strain, for example, each chain is assumed to first reorganise inside its deformed tube, with a Rouse-like dynamics, and then to slowly return to isotropy, relaxing the deformed tube by reptation (see the paper by Montfort et al in this book). Along these lines, the plateau relaxation modulus, the steady state compliance and the zero shear viscosity should be respectively ... 

Although the (Simplex shear modulus is not the most appropriate function to use in all c ses, we wUl describe the linear viscoelastic behaviour in terms of this last function, which is tiie most referred to experimentally furthermore, molecular models are mostly linked to the relaxation modulus, which is the inverse Fourier transform of the complex shear modulus. 

We will discuss in this section the variations of the viscoelastic parameters derived from linear viscoelastic measurements all these parameters may be derived from any t3rpe of measurement (relaxation or creep experiment, mechanical spectroscopy) performed in the relevant time or frequency domain. The discussion will be focused however on the complex shear modulus which is the basic function derived from isothermal frequency sweep measurements performed with modem rotary rheometers. 

The above equation is a one-dimensional model of linear viscoelastic behavior. It can be also written in terms of the relaxation modulus after noting that ... 

One can express linear viscoelasticity using the relaxation spectrum H X), that is, using the relaxation time X. The relationship between the relaxation modulus and the spectra is ... 

In addition, other measurement techniques in the linear viscoelastic range, such as stress relaxation, as well as static tests that determine the modulus are also useful to characterize gels. For food applications, tests that deal with failure, such as the dynamic stress/strain sweep to detect the critical properties at structure failure, the torsional gelometer, and the vane yield stress test that encompasses both small and large strains are very useful. 

The rheological properties of glassy liquids are dominated by one or more very long relaxation times and a high modulus. The detailed linear viscoelastic response varies somewhat with the type of liquid. Some inorganic glassy liquids, such as zinc alkali... 

Oldroyd (1953, 1955) derived expressions for the linear viscoelasticity of suspensions of one Newtonian fluid in another. By using an effective-medium approach, he was able to relax the requirement of diluteness. For an ordinary interface whose interfacial tension r remains constant during the deformation, Oldroyd s result gives the following for the complex modulus G = G + iG" ... 

Accordingly, the phenomenological theory of linear viscoelasticity predicts the same frequency dependence for the loss relaxation modulus of solids and liquids in the terminal region. 

The beauty of the linear viscoelastic analysis lies in the fact that once a viscoelastic function is known, the rest of the functions can be determined. For example, if one measures the comphance function J t), the values of the components of the complex compliance function can in principle be determined from J(t) by using Fourier transforms [Eqs. (6.30)]. On the other hand, the components of the complex relaxation moduh can be obtained from those of / (co) by using Eq. (6.50). Even more, the real components of both the complex relaxation modulus and the complex compliance function can be determined from the respective imaginary components, and vice versa, by using the Kronig-Kramers relations. Moreover, the inverse of the Fourier transform of G (m) and/or G"(co) [/ (co) and/or /"(co)] allows the determination of the shear relaxation modulus (shear creep compliance). Finally, the convolution integrals of Eq. (5.57) allow the determination of J t) and G t) by an efficient method of numerical calculation outlined by Hopkins and Hamming (13). 

It is expected that the same picture that gives a good account of the linear viscoelastic behavior of polymer melts should also hold for semidilute and concentrated solutions. In the case of semidilute solutions some conclusions can be drawn from sealing arguments (19,3, p. 235). In this way, concentration dependence of the maximum relaxation time tmax the zero shear rate viscosity r Q, and the plateau modulus G% can be obtained, where t is the viscosity of the solvent. The relevant parameters needed to obtain Xmax as a function of concentration are b, c, N, kgT, and Dimensional analysis shows that... 

Chapters 5 and 6 discuss how the mechanical characteristics of a material (solid, liquid, or viscoelastic) can be defined by comparing the mean relaxation time and the time scale of both creep and relaxation experiments, in which the transient creep compliance function and the transient relaxation modulus for viscoelastic materials can be determined. These chapters explain how the Boltzmann superposition principle can be applied to predict the evolution of either the deformation or the stress for continuous and discontinuous mechanical histories in linear viscoelasticity. Mathematical relationships between transient compliance functions and transient relaxation moduli are obtained, and interrelations between viscoelastic functions in the time and frequency domains are given. 

Filbey equation (7). For cases of small deformation and deformation gradients, the general linear viscoelastic model can be used for unsteady motion of a viscoelastic fluid. Such a model has a memory function and a relaxation modulus. Bird and co-workers (6, 7) gave details of the available models. 

The contributions of path displacement to linear viscoelastic properties can be obtained using the bond flip modePl The stress relaxation modulus for that model is... 

An example of the linear viscoelastic response in oscillatory shear for a nearly monodisperse linear polybutadiene melt is shown in Fig. 1.2%. Extrapolation of the limiting power laws oiG uP and G" u (the dashed lines in Fig. 7.28) to the point where they cross has special significance. The intersection of the power laws G = J qrj uP and G = t]uj using the above two equations allows us to solve for the frequency where they cross uj= l/(/)/eq), which is the reciprocal of the relaxation time r [Eq. (7.132)]. The modulus level where the two extrapolations cross, obtained by setting a = 1/r = 1 /(/ /eq) iti either equation, is simply the reciprocal of the steady... 

The second important consequence of the relaxation times of all modes having the same temperature dependence is the expectation that it should -bp possible to superimpose linear viscoelastic data taken at different temperatures. This is commonly known as the time-temperature superposition principle. Stress relaxation modulus data at any given temperature Tcan be superimposed on data at a reference temperature Tq using a time scale multiplicative shift factor uj- and a much smaller modulus scale multiplicative shift factor hf. 

The stress relaxation modulus then decays exponentially at the reptation time [Eq. (9.22)]. The terminal relaxation time can be measured quite precisely in linear viscoelastic experiments. Hence, Eq. (9.82) provides the simplest direct means of testing the Doi fluctuation model and evaluating... 

Here a(t,s) is the value of the stress at time t and strain e, E(t) is the relaxation modulus of linear viscoelasticity and /(e) is a function of strain alone. The physical implication of strain-time factorization is that the viscoelastic relaxation processes are independent of strain, a plausible idea for small to moderate deformations. 

Since the linear viscoelasticity of a material is described with a material function G(t), any experiment which gives full information on G(t) is sufficient it is not necessary to give the stresses corresponding to various strain histories. We will restrict the discussion to incompressible isotropic materials. In this case, different types of deformation such as elongation and shear give equivalent information in the range of linear viscoelasticity. Several types of experiments measure relaxation modulus, creep compliance, complex modulus etc which are equivalent to the relaxation modulus (1). 

Thus either G (to) or G"(co) as a function of to gives the information equivalent to that included in G(t) as a function of t. The complex modulus is experimentally a more convenient quantity to describe the linear viscoelasticity of low-viscosity fluids than the relaxation modulus (1). The complex modulus is related to the complex viscosity / (co) and the complex compliance J (to) ... 

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